I'm a beginner in web-mapping. I was playing around with Fusion Tables, but I realized that I had too much data to display.

So here's my question, what are some alternatives to Fusion tables? I have 9 layers of data (one layer is parcel data so there's over 500000 records) and I have to be able to query the data (i.e. how many parcels are within 150m of a school etc.)

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Assuming you are looking for a web mapping with hosting managed for you and various sorts of vector query then one option might be CartoDB. It is backed by a postgres/postgis DB and lets you do various sorts of visualization and mapping without having to setup any services yourself. It includes free options for small tables but you might find you are needing the for-pay levels though even those are quite cheap.

This is a viciously broad question. It depends heavily on what your resources are, and what your goals are. On the simple end, if you're just looking to be able to query GIS data, you could simply load it into a desktop application like QuantumGIS along with a generic basemap (if you need it). On the other hand, If you're trying to build a web application for others to use, you're going to need a database server (preferably PostGIS) with a frontend like GeoServer.